Question

Find the value of
tan⁻¹(1) + cos⁻¹(1/2) + sin⁻¹(1/2)

Answer 1

As we know that range of principal values of

tan⁻¹(x) = [-π/2,π/2]

cos⁻¹(x) = [0,π]

sin⁻¹(x) = [-π/2,π/2]

tan⁻¹(1) = x1

tan x1 = 1 = tan (π/4)

x1 = π/4 belongs to [-π/2,π/2] ...eq1


let
cos⁻¹(1/2) =x2

cos x2 = 1/2 = cos π/3

x2 = π/3 belongs to [0,π]...eq2

let

sin⁻¹(1/2) = x3

x3 = π/6 belongs to [-π/2,π/2]...eq3

at last add all three equations

$\frac{\pi}{4} + \frac{\pi}{3} + \frac{\pi}{6} \\ \\ = \frac{3\pi + 4\pi + 2\pi}{12} \\ \\ = \frac{9\pi}{12} \\ \\ = \frac{3\pi}{4}$

is the final answer